# 25.10: Appendix 25C Analytic Geometric Properties of Ellipses

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Consider Equation (25.3.20), and for now take $$\varepsilon<1$$, so that the equation is that of an ellipse. We shall now show that we can write it as

$\frac{\left(x-x_{0}\right)^{2}}{a^{2}}+\frac{y^{2}}{b^{2}}=1 \nonumber$

where the ellipse has axes parallel to the x - and y -coordinate axes, center at ($$x_{0}$$, 0) , semi-major axis a and semi-minor axis b . We begin by rewriting Equation (25.3.20) as

$x^{2}-\frac{2 \varepsilon r_{0}}{1-\varepsilon^{2}} x+\frac{y^{2}}{1-\varepsilon^{2}}=\frac{r_{0}^{2}}{1-\varepsilon^{2}} \nonumber$

We next complete the square,

$\begin{array}{l} x^{2}-\frac{2 \varepsilon r_{0}}{1-\varepsilon^{2}} x+\frac{\varepsilon^{2} r_{0}^{2}}{\left(1-\varepsilon^{2}\right)^{2}}+\frac{y^{2}}{1-\varepsilon^{2}}=\frac{r_{0}^{2}}{1-\varepsilon^{2}}+\frac{\varepsilon^{2} r_{0}^{2}}{\left(1-\varepsilon^{2}\right)^{2}} \Rightarrow \\ \left(x-\frac{\varepsilon r_{0}}{1-\varepsilon^{2}}\right)^{2}+\frac{y^{2}}{1-\varepsilon^{2}}=\frac{r_{0}^{2}}{\left(1-\varepsilon^{2}\right)^{2}} \Rightarrow \\ \frac{\left(x-\frac{\varepsilon r_{0}}{1-\varepsilon^{2}}\right)^{2}}{\left(r_{0} /\left(1-\varepsilon^{2}\right)\right)^{2}}+\frac{y^{2}}{\left(r_{0} / \sqrt{1-\varepsilon^{2}}\right)^{2}}=1 \end{array} \nonumber$

The last expression in (25.C.3) is the equation of an ellipse with semi-major axis

$a=\frac{r_{0}}{1-\varepsilon^{2}} \nonumber$

semi-minor axis

$b=\frac{r_{0}}{\sqrt{1-\varepsilon^{2}}}=a \sqrt{1-\varepsilon^{2}} \nonumber$

and center at

$x_{0}=\frac{\varepsilon r_{0}}{\left(1-\varepsilon^{2}\right)}=\varepsilon a \nonumber$

as found in Equation (25.B.10).

This page titled 25.10: Appendix 25C Analytic Geometric Properties of Ellipses is shared under a not declared license and was authored, remixed, and/or curated by Peter Dourmashkin (MIT OpenCourseWare) via source content that was edited to the style and standards of the LibreTexts platform; a detailed edit history is available upon request.